Solving Diophantine equations (Q2929240)

The first five chapters of the book belong to general elementary number theory and are disjoint from the subject of Diophantine equations in classical terms. The headings are as follows.NEWLINENEWLINE {\parindent=6mm \begin{itemize}\item[1.] Prime numbers. \item[2.] Smarandache's function \(\eta\). \item[3.] Divisor function \(\sigma\). \item[4.] Euler's totient function \(\varphi\). \item[5.] Generalization of congruence theorems. \item[6.] Analytical solving of Diophantine equations. \item[7.] Partial empirical solving of \(\eta\)-Diophantine equations. NEWLINENEWLINE\end{itemize}} Here a part of authors' philosophy is made clear by stating that a Diophantine equation is any equation for which integer solutions are sought. This is a vast extension of the subject, where traditionally only polynomial equations are allowed, with occasional appearance of exponentials.NEWLINENEWLINETwenty pages are devoted to some mathematical folklore related to linear Diophantine equations. This includes a false statement on page 92 that a linear equation has infinitely many solutions in positive integers if and only if the equation has \textit{variations of sign}, meaning that the gcd of some two coefficients has sign \(-1\). While the latter is meaningless since the greatest common divisor cannot be negative, the essential condition for solvability in integers is needed here: the gcd of the coefficients must divide the free coefficient. The next twenty pages discuss finding integer zeros of a polynomial in one variable and solution of quadratic Diophantine equations with two unknowns. In both cases programs written in Mathcad are presented that search for some numerical solutions.NEWLINENEWLINEThe final Chapter 7 is the principal part of the book. Here \(\eta\)-Diophantine means that the equation involves the \(\eta\)-function. Recall that \(\eta(n)=m\) if \(m\) is the least natural number with the property that \(n\) divides \(m!\). This is called Smarandache function although it was considered already by Lucas in 1883, and others. The examples of \(\eta\)-Diophantine equations considered here are \(\eta(x^m) = x^n\), \(\eta(xy) = \eta(x) \eta(y)\), \(\gcd(x,y) = \gcd(\eta(x),\eta(y))\), etc. There are 27 \(\eta\)-Diophantine equations considered here and for each of them a program in Mathcad is presented for finding some numerical solutions.NEWLINENEWLINEThen the authors proceed to more complicated \(\eta\)-\(s\)-Diophantine equations, where \(s(n) = \sigma(n) - n\). Examples are \(\eta(x) \cdot y =x \cdot s(y)\) and \(\eta(x)^y = s(y)^x\). The next group of equations are \(\eta\)-\(\pi\)-Diophantine equations where \(\pi(n)\) is the number of prime numbers not exceeding \(n\), and then \(\eta\)-\(\sigma_k\)-Diophantine equations and \(\eta\)-\(\varphi\)-Diophantine equations. Altogether there are 62 equations under consideration and for each of them a program in Mathcad is offered for finding some numerical solutions. None of the equations is solved completely.